How this course is built

1. Why a video game and not a video

The meta-analysis by Wouters and colleagues (2013) over 77 studies (more than 5,500 participants) finds that "serious games" produce better learning than traditional instruction (d ≈ 0.29) and above all better retention (d ≈ 0.36). But with one decisive detail: they work better when the game is accompanied by other instruction and when the experience is spread over several sessions.

What we take from it here: no level is "just a game". Every mini-game is followed (or preceded) by a written explanation, a formula taken apart and a worked numerical example. And that is why the Flash review exists: to push towards several sessions instead of one marathon.

2. Why you touch before you read

The interactive simulations of the PhET project (University of Colorado Boulder) show conceptual gains equal to or better than those obtained with real lab apparatus, and they produce more productive discussion. The productive failure line of research (Kapur) further shows that attempting a problem before receiving the explanation — even failing — leads to deeper conceptual understanding than being handed the rule up front.

What we take from it here: in almost every level the slider comes before the formula. First you see the arrows cancel, then we tell you it is called destructive interference.

3. Why there are quizzes (and why they do not give grades)

The testing effect (Roediger & Karpicke) is one of the most solid results in cognitive psychology: recalling a piece of information from memory strengthens it far more than re-reading it for the same amount of time. Reviews of study techniques put practice testing and distributed practice at the top for effectiveness.

What we take from it here: the quizzes are low-stakes, with immediate feedback explaining why, endlessly retryable and with no negative marking. Getting it wrong is part of the method: a question you missed comes back sooner in the review, one you got right comes back later (Leitner box system: 1 day → 3 → 7 → 21 → 60).

4. Why levels only unlock through mastery

In mastery learning (Bloom) the student moves on once they have shown mastery of the current unit, with variable time and attempts instead of fixed ones. The measured effects are among the most consistent in individualised instruction, especially for those starting further behind.

5. Why the interface is so bare

Cognitive load theory (Sweller) and the principles of multimedia learning (Mayer) prescribe some very concrete things, applied here to the letter:

6. Colours, contrast, accessibility

• Dark theme with high-contrast text and line width capped at ~75 characters: two parameters that bear directly on reading fatigue.
• No information carried by colour alone: every state also has a symbol or a label (✓, ✗, 🔒), a WCAG requirement for colour-blind readers.
• Touch targets of at least 44 px, visible focus for keyboard navigation, support for prefers-reduced-motion for those who find animation uncomfortable.
• Immediate feedback on every action (under 100 ms): sliders update the drawing as you move them, not on release. That is the threshold beyond which an interface stops feeling like "yours".

7. Motivation: why XP and levels, but no leaderboards

Self-determination theory (Deci & Ryan) identifies three drivers: competence, autonomy, relatedness. Meta-analyses on gamification show positive but fragile effects: points and badges work when they signal competence gained, and can backfire if they become the only reason to play or if they create social-comparison pressure.

What we take from it here: XP arrives only for things genuinely done (mission passed, question recalled), the ranks describe a skill ("Lord of the arrows", "Period hunter") and there is no leaderboard at all, nor a running timer. Autonomy is protected by free mode and by the ability to replay any level.

8. And the last level?

Research on STEM learning distinguishes between being able to apply and being able to transfer. Transfer is only trained on ill-defined problems, where there is no procedure to repeat. That is why the last level is an open workshop: goals to reach, blocks at your disposal, no solution written down anywhere, and a counter of "how many times you queried the oracle" that pushes you to look for a smarter route.

9. Why the qubit arrives at level 1 and Fourier only at 13

The first version of this course followed the "logical" order: waves → Fourier → qubit. Impeccable on paper, wrong in practice: seven levels of maths before seeing the object you came here for.

The current order follows two well-established principles:

• Spiral curriculum (Bruner, 1960): a concept is met more than once, first in simple form and then in full. Amplitudes appear at level 1 as numbers with a sign (+ and −), which is enough to understand interference and H·H; they become complex arrows at level 8, when two signs are no longer enough. Returning to the same concepts after a gap is also distributed practice: two benefits from one choice.
• Just-in-time: a tool is introduced where it is needed, not before. Waves and the Fourier transform arrive at levels 13–17, right before the QFT, once the question "how do I find a hidden periodicity?" has already occurred to the player — at level 12, with Simon's algorithm, you discover you can only find periods "in XOR", and for real ones you need waves. Introducing them seven levels earlier meant answering a question that had not been born yet.

10. Sound: information, not decoration

An arcade without sound is not an arcade, but noise for the sake of noise violates Mayer's coherence principle (anything that does not serve the concept steals attention). The compromise applied here:

• each type of event has its own timbre, always the same: click, XP, mission, quantum measurement, constructive interference, destructive interference, level fanfare. After a few minutes you recognise what happened without looking: sound becomes a second information channel;
• short sounds (under 350 ms) at low volume, generated live with WebAudio: no file to download, it works on a slow connection too;
• a mistake does not punish: two descending notes, not a buzzer. Punitive feedback raises performance anxiety and reduces willingness to try again, which is exactly what a mastery-based course needs;
• you can turn it all off and the choice is remembered. No sound starts before an interaction, as browser rules also require.

11. "Wow effect": immediate, specific feedback

In the first version, the Part 0 mini-games did not tell you whether you were doing well. Research on feedback (Hattie & Timperley; Shute) is clear on one point: feedback works when it is immediate, specific and about the task, not the person. Hence three additions, now present in all the games:

• The goal always written at the top, in its own band: you do not have to remember what you were asked.
• A "how close are you" bar going from red to amber to green, with a tone that rises as you approach: colour + sound + text, three channels for the same information (also useful to anyone who struggles with colours).
• Visible celebration at the finish line — sparks, flash, sound — because the moment you understood has to be unmistakable. It is the one "decorative" concession, and it lasts half a second.

12. Why an account is mandatory (and why I do not like it)

Asking for registration is friction, and friction loses people: it would be more convenient for everyone to run everything in the browser. Two reasons make it the right choice anyway:

• progress must not die with the cache. A 28-level course is done across several sessions and often on several devices: losing everything to a browser clean-up is the stupidest way to drop out;
• the certificate has to be worth something. The exam questions come from the server without the correct answers and grading happens on the server: if the state lived in the browser, anyone could award themselves 100% with two lines in the console, and the certificate would be worthless paper.

In exchange: only the data that is genuinely needed (first name, last name, email — date of birth is optional and serves to tell apart people with the same name), no profiling, no advertising, total deletion in one click. It is all written in the privacy notice.

13. The AI tutor that refuses to give you the solution

The tutor answers only from the content of this site (RAG) and, by design, does not provide mission solutions: it gives a hint and points you at the right slider. It is an uncomfortable choice but one supported by the research on desirable difficulties (Bjork): the effort of getting there yourself is precisely what produces durable learning. An obliging tutor would make everything look easier and leave less behind.

Every answer cites the level and links to it: the tutor's goal is to put you back inside the game, not to replace it. And if a question finds no answer in the content, it says so instead of making things up.

14. How do I know the simulator is not lying

A course that teaches with a simulator has a basic problem: if the simulator is wrong, it teaches the error — and teaches it convincingly, because it shows it. The project's tests check the properties you would expect (gates stay unitary, the QFT reproduces the Fourier matrix exactly, probabilities sum to one). But those tests are written by me, on the same reasoning the simulator is written on: if the error is in the reasoning, the tests confirm it instead of catching it.

That is why the simulator is compared against an independent implementation: QuantumSim, written in C by Francesco Sisini. Three hundred randomly generated circuits — up to 4 qubits, with Hadamard, Pauli, S, T, T†, phase rotations, CNOT, CZ and Toffoli — are run through both simulators and the amplitudes compared one by one. Different languages, different authors, code written without knowing each other: the largest gap is of the order of 10⁻¹⁵, that is, the limit of the computer's number precision. An error common to both, at that point, is very unlikely.

QuantumSim is released under the GNU GPL v3 and is not included in this site: it is downloaded and compiled only when the check is run (npm run test:incrociato), like a bench tool. Thanks to Francesco Sisini for letting me use it — and above all because it is from his books that I started learning this subject.

15. The site's copy: which persuasion techniques I use, declared

This site also has a professional purpose: making me known as someone who builds AI systems and teaches. I think it is only fair to declare which levers I am pulling, so you can weigh them:

16. How it is written for search engines and for AIs

Half of today's searches end up inside an AI-generated answer instead of a list of links. The guidance emerging from research on Generative Engine Optimization is consistent with writing well:

• structured data (JSON-LD: Course, Person, FAQPage) so a machine can tell what this resource is, who wrote it and what it teaches;
• self-contained, quotable statements: every important answer sits in a paragraph that stands on its own, without needing the surrounding context;
• an llms.txt file with the project's verifiable facts, for anyone indexing with language models;
• trust signals: author in the open, linked sources, update date, and explicit admission of the limits (the certificate is not accredited).

Note: the same research shows that the content cited by AIs is the structured and verifiable kind. Another way of saying that writing honestly and writing to be found, for once, coincide.

17. The three languages, and how you move between them

The course exists in full in Italian, English and Spanish: not a translated summary, three complete editions — exam and certificate included. The addresses are translated too (/en/lessons/, /es/lecciones/), because a page in Spanish living in a folder called "lezioni" is a half-translated page, and it shows.

The selector at the top follows four rules, and none of them is about looks:

• Every language is written in its own language — «Español», not «Spanish». People look for their own language the way it is written at home, not translated into a language they may not read. It is the recommendation of the W3C and of the Nielsen Norman Group, and it is also why the first version did not work: it said «IT EN ES», and ISO codes are mute labels to anyone who has never seen them.
• No flags. A flag stands for a state, not a language: which one would you put on Spanish, out of the twenty countries that speak it? In their place a globe, the one symbol the public associates with "language" without associating it with a country.
• You change language without losing your place. Someone reading the QFT in Italian who picks English wants the QFT in English, not to start over from the map. The addresses of the other versions are the same hreflang links the page already declares for search engines: computing them a second time would be the classic way of letting them drift apart.
• If your browser speaks another language I say so, but I do not move you. A single line appears — written in the language it offers, otherwise it cannot be read — with two buttons: switch, or stay. Then it never asks again. Automatic redirection is explicitly discouraged by Google for multilingual sites: it stops you reaching a particular version on purpose, it confuses people who speak more languages than their browser is configured for, and it hides the other copies from search engines.

Sources

Listed in the order they appear in the text. Where there is no link it is because the reference is a book or a classic paper that is easy to find: I prefer citing it that way rather than linking a copy of dubious provenance.

• Wouters, P., van Nimwegen, C., van Oostendorp, H., van der Spek, E. (2013). A meta-analysis of the cognitive and motivational effects of serious games. Journal of Educational Psychology, 105(2), 249–265. — ERIC record
• Clark, D. B., Tanner-Smith, E. E., Killingsworth, S. S. (2016). Digital Games, Design, and Learning: A Systematic Review and Meta-Analysis. Review of Educational Research, 86(1), 79–122. — paper
• Sailer, M., Homner, L. (2020). The Gamification of Learning: a Meta-analysis. Educational Psychology Review. — paper
• Perkins, K. et al. PhET: Interactive Simulations for Teaching and Learning Physics. The Physics Teacher, 44(1), 18–23. — paper
• Kapur, M. Productive Failure. — accessible summary (PDF)
• Roediger, H. L., Karpicke, J. D. — on the testing effect and the forward effect of recall: Retrieval practice enhances new learning (PMC)
• Systematic review on spaced learning, interleaving and retrieval practice (2023), Journal of the American College of Radiology. — paper
• Accessible summary on recall + spaced practice: Evidence Based Education
• Bjork, R. A. — Desirable Difficulties in Theory and Practice: why what makes learning harder in the moment makes it more solid over time. — text
• Kivetz, R., Urminsky, O., Zheng, Y. (2006). The Goal-Gradient Hypothesis Resurrected. Journal of Marketing Research. — Columbia Business School summary
• Nunes, J. C., Drèze, X. (2006). The Endowed Progress Effect: How Artificial Advancement Increases Effort. Journal of Consumer Research. — text
• Sisini, F. — QuantumSim, a quantum circuit simulator in C, and the popular books by the same author from which my path into this subject began. — repository
• Bruner, J. S. (1960). The Process of Education — the spiral curriculum: returning to the same concepts at ever deeper levels.
• Sweller, J. — cognitive load theory; Mayer, R. E. — principles of multimedia learning (coherence, signalling, contiguity, segmenting, pre-training, conversational style).
• Hattie, J., Timperley, H. (2007). The Power of Feedback; Shute, V. (2008). Focus on Formative Feedback — feedback works if it is immediate, specific and task-related.
• Deci, E. L., Ryan, R. M. — self-determination theory: competence, autonomy, relatedness.
• Cialdini, R. B. — Influence: reciprocity, social proof, authority. Used here only on verifiable claims.
• On Generative Engine Optimization: overview of practices and position paper on the risks (arXiv)
• WCAG 2.2 (W3C) for contrast, touch targets, visible focus and information never carried by colour alone. — quick reference

The whole project is public on GitHub under a free non-commercial licence: studying it, modifying it and using it in state education is always allowed; paid training needs an agreement. If you find an error in these pages, report it.